Representation of numbers with a single 4
Representing numbers as a result of arithmetic operations on a restricted set of numbers is an entertaining activity (Three 3's, Three 4's, Three 5's, Four 3's, Four 4's, Four 5's) that even lends itself to some degree of systematization. Since the basic arithmetic operations take two arguments (a+b, a*b, a^{b}, etc.), it never occurred to me to consider representing numbers with a single selected number. For instance, 1 = [3], 2 = [3!], 3 = 3. However, I received the following letter from one of the visitors:
I have been interested in this puzzle for thirty years. Is it possible to represent the numbers 1  12 mathematically only using 1 four?
Regards, Robert Smith Well, there are at least two ways to look at the table. For one, I'll use it as an index to introduce the number families referred to by Robert (Fibonacci numbers, Bell numbers, ...). Secondly, since accepting notations that index such families of numbers (e.g., F_{4}) would mean going beyond the original problem of representing numbers with arithmetic operations, I consider this a challenge to put up a similar table while staying in the framework of arithmetic operations. Below are the results of my first attempt. You are welcome to suggest more entries. I would still allow for supefactorial and subfactorial notations, although they are far from being standard. To shorten the representations, I shall use the conventions introduced for the same purpose elsewhere. Thus {n} refers to an entry in row #n provided, of course, it has already been filled. (exp(x) = e^{x} and its inverse ln(x) are being used somewhat reluctantly.)
^{(1)} By Don Gosiewski ^{(2)} By Andre Gustavo dos Santos ^{(3)} By Don Double Factorialn! is the product of all integers less or equal to n that have the same parity as n. 7!! = 7*5*3*1, 10!! = 10*8*6*4*2. Superfactorialn$ is the product of all factorials k! of integers not exceeding n: Subfactorial!n is the number of Derangements (i.e., permutations in which no number appears in its original place.) There are several ways to define this number. One relates it to the factorial: The number was discovered independently by Niclaus Bernoulli and L.Euler. The idea of derangements is exemplified in a concrete form by the following problem of misaddressed letters:
